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SageMath
E = EllipticCurve("fb1")
E.isogeny_class()
Elliptic curves in class 324870fb
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
324870.fb4 | 324870fb1 | \([1, 0, 0, 77419, -1399455]\) | \(436192097814719/259683840000\) | \(-30551544092160000\) | \([2]\) | \(3317760\) | \(1.8525\) | \(\Gamma_0(N)\)-optimal |
324870.fb3 | 324870fb2 | \([1, 0, 0, -314581, -11356255]\) | \(29263955267177281/16463793153600\) | \(1936948800727886400\) | \([2, 2]\) | \(6635520\) | \(2.1991\) | |
324870.fb2 | 324870fb3 | \([1, 0, 0, -3146781, 2137150665]\) | \(29291056630578924481/175463302795560\) | \(20643082110594838440\) | \([2]\) | \(13271040\) | \(2.5457\) | |
324870.fb1 | 324870fb4 | \([1, 0, 0, -3754381, -2795530375]\) | \(49745123032831462081/97939634471640\) | \(11522500055953974360\) | \([2]\) | \(13271040\) | \(2.5457\) |
Rank
sage: E.rank()
The elliptic curves in class 324870fb have rank \(1\).
Complex multiplication
The elliptic curves in class 324870fb do not have complex multiplication.Modular form 324870.2.a.fb
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.